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  1. Thermal cConductivity of sSingle lLayer MoS2
  2. Abstract
  3. Abbreviations
  4. 1INTRODUCTION
  5. 1.1Background
  6. 1.2 Objectives of the Research
  7. 2LITERATURE REVIEW
  8. 2.1The Force Field
  9. 3METHODOLOGY
  10. 3.1Concepts
  11. 3.2Methods
  12. 4RESULTS AND DISCUSSION
  13. 5CONCLUSION
  14. ACKNOWLEDGEMENTS
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Thermal conductivity of single layer MoS2

Nisarg Vyas

School of Physical Sciences, National Institute of Science Education and Research, Jatni, Odisha 752050

Prof. Prabal Maiti

Department of Physics, Indian Institute of Science, Bengaluru 560012

Abstract

A popular representative of transition metal dichalcogenides, MoS2 is taken up for studying the thermal properties and their dependence on temperature. The material is considered only in its 2-dimensional form with a single layer. The atomic interactions are modelled using Stillinger-Weber empirical type potential. We compute values for the thermal conductivity of mono-layer MoS2 via Green-Kubo formalism based equilibrium molecular dynamics method. The values thus obtained, match with other computational works. In the presented work, it is also verified that the thermal conductivity of the considered material scales inversely with the system temperature. All of the molecular dynamics simulations are performed in large-scale atomic/molecular massively parallel simulator (LAMMPS).

Keywords: transition metal dichalcogenides, thermal conductivity, equilibrium molecular dynamics, Green-Kubo, Stillinger-Weber, LAMMPS

Abbreviations

Abbreviations
TMDCTransition Metal Dichalcogenides
SWStillinger-Weber
LAMMPSLarge-scale Atomic/Molecular Massively Parallel Simulator
EMDEquilibrium Molecular Dynamics

INTRODUCTION

Background

Graphene, owing to its excellent thermal conductivity and high intrinsic mobility, has been a toy tool of theoretical physicists and has been studied extensively in recent years. The peculiar way in which the thermal conductivity of graphene scales as one goes from bulk to few-layer and to mono-layer graphene also makes it an interesting object to study thermal properties of matter and to dwell deeper into the mechanisms of heat conduction phenomena. However, due to the lack of electronic band gap in graphene, a very limited number of applications in nano electronics have been possible.

To overcome this deficiency, researchers have shifted their attention to another class of 2-dimensional materials: transition metal dichalcogenides (TMDCs). These materials are semiconducting in nature and can be represented as MX2 where M can be a transition metal (like Mo, W etc.) and X can be a chalcogen (S, Se or Te). These semiconductors possess rich electronic and photonic properties with promises for novel applications in nanoelectronics. One of the applications of TMDCs is in thermo electronics where strict control over thermal properties is required. For effective implementation of the materials, it is necessary to gain a better understanding of thermal transport in them.

Objectives of the Research

In literature, the computationally predicted thermal conductivity values for single layer MoS2 vary from 1.35-5.8 W/(m.K) [​Liu, et al, 2013​] to 103 W/(m.K) [​Gu, et al, 2014​]. The largely underestimated value obtained earlier by Liu et al, were due to the force field used. However the later studies done by Li, et al, 2013 using ab-initio methods and Boltzmann transport equations give an estimate of 83 W/(m.K). This is quite close to the experimentally obtained values of 84 ± 17 W/(m.K), achieved by Zhang, et al, 2015 using opto-thermal Raman technique. The objectives taken up in the currently undertaken study is to calculate the in-plane thermal conductivity value for mono-layer MoS2 using SW potential and compare the results with the existing experimental values and other simulations done either with the same potential or different ones. The dependence of thermal conductivity on temperature is also studied between 50 and 450 K.

LITERATURE REVIEW

The Force Field

To properly account for the structural, vibrational and mechanical properties of the material, the Stillinger-Weber (SW) potential developed by Cem Sevik, 2016 is used. It has been shown that the interatomic interactions described by SW type empirical potentials give thermal properties that match the experimentally calculated valuues [Yan, et al, 2014]. It has also been noted that the parameters produced in this potential are in accordance with results from ab-initio methods of density functional theory for both MoS2 and MoSe2 [Gu, et al, 2014].

The SW potential as proposed in 1985 [Stillinger, et al, 1985] defines on site potential as:

Ui = 12i jV2(rij) + 12j i k i, jhijk

Where V2 describes the two body terms called the bond-stretching terms, while hijk defines three body terms appropriately called bond-bending terms. Note that two-index terms work for a pair of atoms while three-index quantities are detrimental for triplets of atoms. The individual potential form can be given in terms of material specific parameters Aij , Bij , εij, σij, aij, λijk, γijand cosθijk as follows.

V2(rij) = AijεijBijσijrij4-1 exp1rijσij-aij

hijk = εijλijk expγijrijσij-aij + γikrikσik-aik×cosθijk - cosθ0ijk2

METHODOLOGY

Concepts

There are quite a few well proven methods in molecular dynamics to calculate thermal conductivity. Among these are the equilibrium molecular dynamics (EMD) method based on Green-Kubo relation and Fourier law based other non-equilibrium molecular dynamics (NEMD) methods. When the heat transport in diffusive, both of the methods give consistent calculations on thermal properties. However we use the Green-Kubo formalism in the present work to estimate the thermal conductivity of mono-layer MoS2.

The Green-Kubo formalism works as follows. The ensemble average of heat current auto-correlation function is calculated and then is numerically integrated to obtain a value of thermal conductivity. The formula which relates the ensemble averaged quantity with, thermal conductivity, kappa is given as follows:

κ  =  VkBT20Jx(0)Jx(t)dt   =  V3kBT20J(0).J(t)dt

where kB is the Boltzmann's constant, T is the system temperature and V is the system volume. Note that the above two expressions equate only when isotropic bulk systems are considered. In the present case of 2-D MoS2 sheet, it being anisotropic, only the x component of heat flux is considered in calculations to give a thermal conductivity value along the length (in the direction of the x axis). The heat current J is calculated as

J = 1Vieivi - iSivi

In the first term, ei are the energy terms (both potential and kinetic) while Vi is a 3×3 matrix multiplied to energy vector to get a vector again. The Si is per atom stress tensor. Both of these quantites are computes at every timestep by appropriate compute commands in LAMMPS.

Methods

The system used in the study is a 11.2 nm long and 2.2 nm wide ribbon of mono-layer MoS2. The out of plane direction is the direction of the z axis while the length of the ribbon is along the x axis (see fig. 1 and 2). The simulation box has periodic boundary conditions in the direction of the y and z axes while the box’s boundaries are non-periodic in nature along the length. However, the effect of periodic boundaries along the z axis is nullified by keeping sufficient vacuum on both sides of the ribbon. The initial structure prepared with approximate interatomic distances is minimized for the potential energy using the conjugate gradient algorithm implemented in LAMMPS. After the minimization step, the correctness of the MoS2 structure is determined by observing the radial distribution function with OVITO, a visualization software. Peaks seen in the radial distribution function, which shows the variation of density as a function of distance from a particle, appear at values that correspond to the nearest and second nearest neighbour distance for MoS2.

Screenshot from 2019-07-25 10-09-20_1.png
    Out of plane or top view of the MoS2 ribbon. White spheres are Sulphur atoms while orange ones are Molybdenum atom.
    Screenshot from 2019-07-25 10-11-23.png
    Edge view along the y axis (along the width)
      Screenshot from 2019-07-25 10-07-44_1.png
      Side view along the x axis (along the length). Note that the sheet ends in two different ways with Mo on one side and with S atoms on the other.
        Structure of MoS2 sheet used in the simulation. Visualization in OVITO

        The protocol followed throughout the simulations is as follows. The atoms are given a Maxwellian velocity distribution in accordance with the desired temperature for the system. The prepared structure is then kept in a canonical ensemble (nvt) for 500 ps to equilibrate the temperature of the system. Once the system reaches a steady state, the heat current at every 10th step is used for calculation for 5×106 steps. This gives a numerical integration of the heat current auto-correlation function over a total time period of 5 ns. While the integral is evaluated, the system simultaneously remains in a micro-canonical ensemble (nve). Note that the Green-Kubo method is called EMD because the heat current is calculated when the system resides in a steady equilibrium state, this justifies the invoking of micro-canonical ensemble.

        convergence.png
          On x axis is the number of simulation steps versus thermal conductivity values to show convergence.

          The kappa calculation and averaging is done only after making sure that the auto-correlated flux values have converged. See fig. 3 for similar check made on the calculation for system at 300 K. The volume of the system, needed to calculate the thermal conductivity using equation in ​Sec. 3.1​, is computed using surface mesh analysis in OVITO which computes the solid volume after constructing a closed geometrical surface enclosing all the atoms within it [Stukowski, et al, 2013]. For simple systems like the one considered here, the method is doubtlessly accurate.

          RESULTS AND DISCUSSION

          After equilibrating the system at different temperatures, each time the thermal conductivity calculations are averaged for the last 1 ns of the simulation time. The average value when plotted against temperature with the standard deviation of the data as the error bar, gives the following plot (fig. 4).

          kappa_vs_T.png
            The fit obtained for k vs. T; f(x) = 84.25*(1/T) + 6.89

            The thermal conducivity estimation, for a MoS2 sheet of approximate length 11 nm, at 300 K is around 7.1 W/(m.K) which lies close to 5 W/(m.K), an estimation given by Hong, et al, 2016 at the same length and temperature and using the same force fields. However their method of estimation is Green-Kubo as well as non-equilibrium molecular dynamics method based on Fourier's law. The 1/T curve fitting with the corresponding kappa values has a Chi square value of 0.3.

            CONCLUSION

            The obtained values of thermal conductivity for single layer MoS2 are close to those obtained by Hong, et al, 2016. However, it will be too hasty to deduce the inverse dependence of kappa on the system temperature as the plot does not have satisfactory number of data points. We realize that the claim that kappa depends inversely on temperature can be strengthened only by generating statistically enough data points. Increasing the number of simulation runs per data point would not only give more reliable thermal conductivity values but would also prevent the underestimation of errors.

            Nevertheless, the effort to compute and cross check the thermal conductivity of single layer MoS2 can be safely said to have been accomplished. Further studies in this direction can be taken up with confidence. Similar to the heterostructure proposed by Yang, et al, 2012, it has not escaped our notice that a heterostructure of MoS2 and MoSe2 can be exploited to achieve a material of thermal conductivity lower than both of the materials individually. This can be taken up as a further study steming from the one done here.

            ACKNOWLEDGEMENTS

            No man is an island. In this journey of about two months I have been imensely fortunate to work and interact with good people who helped me in various ways and by various means. Thanks to Vignesh for being a strict mentor. This work would not have completed without you. I would like to extend heartiest thanks to Shubhadeep Dasgupta and Jeet Majumdar for their constant support and for the insights they provided into methods of molecular dynamics, without which this work would not have made much sense to me. A sincere thanks goes to Indrajit Maity for helping me with his expertize whenever I was in trouble. A cheerful thanks to my cousins, Jayaswini and Sumedha, and to my uncle and aunt for making my stay in Bangalore so comfortable and for making my weekends enjoyful!

            Lastly, I wish to express my gratitude towards Prof. Prabal K Maiti and the Indian Academy of Sciences for providing me this opportunity to learn and to grow.

            References

            • Liu, Xiangjun and Zhang, Gang and Pei, Qing-Xiang and Zhang, Yong-Wei (2013). Phonon thermal conductivity of monolayer MoS2 sheet and nanoribbons. 103,

            • Gu, Xiaokun and Yang, Ronggui (2014). Phonon transport in single-layer transition metal dichalcogenides: A first-principles study. 105,

            • Li, Wu and Carrete, J. and Mingo, Natalio (2013). Thermal conductivity and phonon linewidths of monolayer MoS2 from first principles. 103,

            • Zhang, Xian and Sun, Dezheng and Li, Yilei and Lee, Gwan-Hyoung and Cui, Xu and Chenet, Daniel and You, Yumeng and Heinz, Tony F. and Hone, James C. (2015). Measurement of Lateral and Interfacial Thermal Conductivity of Single- and Bilayer MoS2 and MoSe2 Using Refined Optothermal Raman Technique. 7,

            • Ali Kandemir, Haluk Yapicioglu, Alper Kinaci, Tahir Çağın, Cem Sevik, 2016, Thermal transport properties of MoS2and MoSe2monolayers, Nanotechnology, vol. 27, no. 5, pp. 055703

            • Yan, Rusen and Simpson, Jeffrey R. and Bertolazzi, Simone and Brivio, Jacopo and Watson, Michael and Wu, Xufei and Kis, Andras and Luo, Tengfei and Hight Walker, Angela R. and Xing, Huili Grace (2014). Thermal Conductivity of Monolayer Molybdenum Disulfide Obtained from Temperature-Dependent Raman Spectroscopy. 8,

            • Stillinger, Frank H. and Weber, Thomas A. (1985). Computer simulation of local order in condensed phases of silicon. 31,

            • Stukowski, Alexander (2013). Computational Analysis Methods in Atomistic Modeling of Crystals. 66,

            • Hong, Yang and Zhang, Jingchao and Zeng, Xiao Cheng (2016). Thermal Conductivity of Monolayer MoSe2 and MoS2. 120,

            • Yang, Kaike and Chen, Yuanping and D’Agosta, Roberto and Xie, Yuee and Zhong, Jianxin and Rubio, Angel (2012). Enhanced thermoelectric properties in hybrid graphene/boron nitride nanoribbons. 86,

            Source

            • Fig 1: Visualization in OVITO
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            • 0 Thermal conductivity of single layer MoS2
            • 0 Abstract
            • 0 Abbreviations
            • 0 1 INTRODUCTION
            • 0 1.1 Background
            • 0 1.2 Objectives of the Research
            • 0 2 LITERATURE REVIEW
            • 0 2.1 The Force Field
            • 0 3 METHODOLOGY
            • 1 3.1 Concepts
            • 0 3.2 Methods
            • 0 4 RESULTS AND DISCUSSION
            • 0 5 CONCLUSION
            • 0 ACKNOWLEDGEMENTS